Outline
Week 9: complex numbers; complex exponential and polar form
Course Notes: 5.1, 5.2, 5.3, 5.4
Goals:
Fluency with arithmetic on complex numbers
Using matrices with complex entries: finding determinants and inverses,
solving systems, etc.
Visualizing complex numbers in coordinate systems
Complex Arithmetic
i
We use i (as in ”imaginary”) to denote the number whose square is −1.
Complex Arithmetic
i
We use i (as in ”imaginary”) to denote the number whose square is −1.
i 2 = −1
Complex Arithmetic
i
We use i (as in ”imaginary”) to denote the number whose square is −1.
i 2 = −1
(−i)2 =
Complex Arithmetic
i
We use i (as in ”imaginary”) to denote the number whose square is −1.
i 2 = −1
(−i)2 = − 1
Complex Arithmetic
i
We use i (as in ”imaginary”) to denote the number whose square is −1.
i 2 = −1
(−i)2 = − 1
i3 =
Complex Arithmetic
i
We use i (as in ”imaginary”) to denote the number whose square is −1.
i 2 = −1
(−i)2 = − 1
i 3 =−i
Complex Arithmetic
i
We use i (as in ”imaginary”) to denote the number whose square is −1.
i 2 = −1
(−i)2 = − 1
i 3 =−i
i4 =
Complex Arithmetic
i
We use i (as in ”imaginary”) to denote the number whose square is −1.
i 2 = −1
(−i)2 = − 1
i 3 =−i
i 4 =1
Complex Arithmetic
i
We use i (as in ”imaginary”) to denote the number whose square is −1.
(−i)2 = − 1
i 3 =−i
i 4 =1
i 2 = −1
When we talk about ”complex numbers,” we allow numbers to have real
parts and imaginary parts:
2 + 3i
−1
2i
Complex Arithmetic
2 + 3i
−1
2i
imaginary
real
Complex Arithmetic
−1
2 + 3i
2i
imaginary
2 + 3i
3
real
2
Complex Arithmetic
−1
2 + 3i
2i
imaginary
2 + 3i
3
−1
real
2
Complex Arithmetic
−1
2 + 3i
2i
imaginary
2 + 3i
3
2i
−1
real
2
Complex Arithmetic
Addition happens component-wise, just like with vectors or polynomials.
imaginary
real
Complex Arithmetic
Addition happens component-wise, just like with vectors or polynomials.
(2 + 3i) + (3 − 4i) =
imaginary
2 + 3i
real
3 − 4i
Complex Arithmetic
Addition happens component-wise, just like with vectors or polynomials.
(2 + 3i) + (3 − 4i) = 5 − i
imaginary
2 + 3i
real
5−i
3 − 4i
Complex Arithmetic
Multiplication is similar to polynomials.
Complex Arithmetic
Multiplication is similar to polynomials.
(2 + 3i)(3 − 4i)=
Complex Arithmetic
Multiplication is similar to polynomials.
(2 + 3i)(3 − 4i)=2 · 3 + 3i · 3 + (2)(−4i) + (3i)(−4i)
Complex Arithmetic
Multiplication is similar to polynomials.
(2 + 3i)(3 − 4i)=2 · 3 + 3i · 3 + (2)(−4i) + (3i)(−4i)
= 6 + 9i − 8i + 12
Complex Arithmetic
Multiplication is similar to polynomials.
(2 + 3i)(3 − 4i)=2 · 3 + 3i · 3 + (2)(−4i) + (3i)(−4i)
= 6 + 9i − 8i + 12= 18 + i
Complex Arithmetic
Multiplication is similar to polynomials.
(2 + 3i)(3 − 4i)=2 · 3 + 3i · 3 + (2)(−4i) + (3i)(−4i)
= 6 + 9i − 8i + 12= 18 + i
I: 0
A: (−4 + 3i) + (1 − i)
B: i(2 + 3i)
II: -1
III: -2
IV: 2i+12
C: (i + 1)(i − 1)
D: (2i + 3)(i + 4)
V: -3+2i
VI: 3+2i
VII: 10+11i
Complex Arithmetic
Multiplication is similar to polynomials.
(2 + 3i)(3 − 4i)=2 · 3 + 3i · 3 + (2)(−4i) + (3i)(−4i)
= 6 + 9i − 8i + 12= 18 + i
I: 0
A: (−4 + 3i) + (1 − i)
B: i(2 + 3i)
II: -1
III: -2
IV: 2i+12
C: (i + 1)(i − 1)
D: (2i + 3)(i + 4)
V: -3+2i
VI: 3+2i
VII: 10+11i
Complex Arithmetic
Modulus
The modulus of (x + yi) is:
|x + yi| =
like the norm of a vector.
p
x2 + y2
Complex Arithmetic
Modulus
The modulus of (x + yi) is:
|x + yi| =
p
x2 + y2
like the norm of a vector.
Complex Conjugate
The complex conjugate of (x + yi) is:
x + yi = x − yi
the reflection of the vector over the real (x) axis.
Complex Arithmetic
|x + yi| =
p
x2 + y2
x + yi = x − yi
Complex Arithmetic
|x + yi| =
p
x2 + y2
x + yi = x − yi
Suppose z = x + yi and w = a + bi. Calculate the following.
• z
• |z|
• |z|
• z −z
• z +z
• zz − |z|2
• zw − (z)(w )
Complex Arithmetic
|x + yi| =
p
x2 + y2
x + yi = x − yi
Suppose z = x + yi and w = a + bi. Calculate the following.
• z
• |z|
• |z|
y is called the imaginary part of z
• z − z = 2yi
• z +z
• zz − |z|2
• zw − (z)(w )
Complex Arithmetic
|x + yi| =
p
x2 + y2
x + yi = x − yi
Suppose z = x + yi and w = a + bi. Calculate the following.
• z
• |z|
• |z|
y is called the imaginary part of z
• z − z = 2yi
x is called the real part of z
• z + z= 2x
• zz − |z|2
• zw − (z)(w )
Complex Arithmetic
|x + yi| =
p
x2 + y2
x + yi = x − yi
Suppose z = x + yi and w = a + bi. Calculate the following.
• z
• |z|
• |z|
y is called the imaginary part of z
• z − z = 2yi
x is called the real part of z
• z + z= 2x
So, zz = |z|2
• zz − |z|2 = 0
• zw − (z)(w )
Complex Arithmetic
|x + yi| =
p
x2 + y2
x + yi = x − yi
Suppose z = x + yi and w = a + bi. Calculate the following.
• z
• |z|
• |z|
y is called the imaginary part of z
• z − z = 2yi
x is called the real part of z
• z + z= 2x
So, zz = |z|2
• zz − |z|2 = 0
So, zw = z w
• zw − (z)(w )= 0
Complex Arithmetic
|x + yi| =
p
x2 + y2
x + yi = x − yi
Suppose z = x + yi and w = a + bi. Calculate the following.
• z
• |z|
• |z|
y is called the imaginary part of z
• z − z = 2yi
x is called the real part of z
• z + z= 2x
So, zz = |z|2
• zz − |z|2 = 0
So, zw = z w
• zw − (z)(w )= 0
Division
z
=
w
Complex Arithmetic
|x + yi| =
p
x2 + y2
x + yi = x − yi
Suppose z = x + yi and w = a + bi. Calculate the following.
• z
• |z|
• |z|
y is called the imaginary part of z
• z − z = 2yi
x is called the real part of z
• z + z= 2x
So, zz = |z|2
• zz − |z|2 = 0
So, zw = z w
• zw − (z)(w )= 0
Division
z
z w
=
·
w
w w
Complex Arithmetic
|x + yi| =
p
x2 + y2
x + yi = x − yi
Suppose z = x + yi and w = a + bi. Calculate the following.
• z
• |z|
• |z|
y is called the imaginary part of z
• z − z = 2yi
x is called the real part of z
• z + z= 2x
So, zz = |z|2
• zz − |z|2 = 0
So, zw = z w
• zw − (z)(w )= 0
Division
z
z w
zw
=
·
=
w
w w
|w |2
Complex Arithmetic
z
zw
=
w
|w |2
Complex Arithmetic
z
zw
=
w
|w |2
Compute:
•
2+3i
3+4i
•
1+3i
1−3i
•
2
1+i
•
5
i
Complex Arithmetic
z
zw
=
w
|w |2
Compute:
•
2+3i
3+4i
•
1+3i
1−3i
•
2
1+i
•
5
i
=
18
25
+
1
25 i
Complex Arithmetic
z
zw
=
w
|w |2
Compute:
•
2+3i
3+4i
=
18
25
+
•
1+3i
1−3i
=
−4
5
+ 35 i
•
2
1+i
•
5
i
1
25 i
Complex Arithmetic
z
zw
=
w
|w |2
Compute:
•
2+3i
3+4i
=
18
25
+
•
1+3i
1−3i
=
−4
5
+ 35 i
•
2
1+i
•
5
i
=1−i
1
25 i
Complex Arithmetic
z
zw
=
w
|w |2
Compute:
•
2+3i
3+4i
=
18
25
+
•
1+3i
1−3i
=
−4
5
+ 35 i
•
2
1+i
•
5
i
=1−i
= −5i
1
25 i
Polynomial Factorizations
Theorem
Every polynomial can be factored completely over the complex numbers.
Polynomial Factorizations
Theorem
Every polynomial can be factored completely over the complex numbers.
Example: x 2 + 1 = (x − i)(x + i)
Polynomial Factorizations
Theorem
Every polynomial can be factored completely over the complex numbers.
Example: x 2 + 1 = (x − i)(x + i)
Example: x 2 + 2x + 10 =
Polynomial Factorizations
Theorem
Every polynomial can be factored completely over the complex numbers.
Example: x 2 + 1 = (x − i)(x + i)
Example: x 2 + 2x + 10 = (x + 1 + 3i)(x + 1 − 3i)
Polynomial Factorizations
Theorem
Every polynomial can be factored completely over the complex numbers.
Example: x 2 + 1 = (x − i)(x + i)
Example: x 2 + 2x + 10 = (x + 1 + 3i)(x + 1 − 3i)
Example: x 2 + 4x + 5 =
Polynomial Factorizations
Theorem
Every polynomial can be factored completely over the complex numbers.
Example: x 2 + 1 = (x − i)(x + i)
Example: x 2 + 2x + 10 = (x + 1 + 3i)(x + 1 − 3i)
Example: x 2 + 4x + 5 =(x + 2 + i)(x + 2 − i)
Calculating Determinants
We calcuate the determinant of a matrix with complex entries in the same
way we calculate the determinant of a matrix with real entries.
Calculating Determinants
We calcuate the determinant of a matrix with complex entries in the same
way we calculate the determinant of a matrix with real entries.
1+i 1−i
det
2
i
Calculating Determinants
We calcuate the determinant of a matrix with complex entries in the same
way we calculate the determinant of a matrix with real entries.
1+i 1−i
det
= (1 + i)(i) − (1 − i)(2) =
2
i
Calculating Determinants
We calcuate the determinant of a matrix with complex entries in the same
way we calculate the determinant of a matrix with real entries.
1+i 1−i
det
= (1 + i)(i) − (1 − i)(2) = − 3 + 3i
2
i
Gaussian Elimination
Give a parametric equation for all solutions to the homogeneous system:
ix1 +
2ix1
x2 + 2x3 = 0
ix2 + 3x3 = 0
+ (2 − i)x2 + x3 = 0
Gaussian Elimination
Give a parametric equation for all solutions to the homogeneous system:
ix1 +
2ix1
x2 + 2x3 = 0
ix2 + 3x3 = 0
+ (2 − i)x2 + x3 = 0
Solve the following system of equations:
ix1 +
2x2 =
9
3x1 + (1 + i)x2 = 5 + 8i
Gaussian Elimination
Give a parametric equation for all solutions to the homogeneous system:
ix1 +
2ix1
x2 + 2x3 = 0
ix2 + 3x3 = 0
+ (2 − i)x2 + x3 = 0
Solve the following system of equations:
ix1 +
2x2 =
9
3x1 + (1 + i)x2 = 5 + 8i
Find the inverse of the matrix
i 1
2 3i
Gaussian Elimination
Give a parametric equation for all solutions to the homogeneous system:
ix1 +
2ix1
x2 + 2x3 = 0
ix2 + 3x3 = 0
+ (2 − i)x2 + x3 = 0
[x1 , x2 , x3 ] = s[−3 + 2i, 3i, 1]
Solve the following system of equations:
ix1 +
2x2 =
9
3x1 + (1 + i)x2 = 5 + 8i
Find the inverse of the matrix
i 1
2 3i
Gaussian Elimination
Give a parametric equation for all solutions to the homogeneous system:
ix1 +
2ix1
x2 + 2x3 = 0
ix2 + 3x3 = 0
+ (2 − i)x2 + x3 = 0
[x1 , x2 , x3 ] = s[−3 + 2i, 3i, 1]
Solve the following system of equations:
ix1 +
2x2 =
9
3x1 + (1 + i)x2 = 5 + 8i
x1 = i, x2 = 5
Find the inverse of the matrix
i 1
2 3i
Gaussian Elimination
Give a parametric equation for all solutions to the homogeneous system:
ix1 +
2ix1
x2 + 2x3 = 0
ix2 + 3x3 = 0
+ (2 − i)x2 + x3 = 0
[x1 , x2 , x3 ] = s[−3 + 2i, 3i, 1]
Solve the following system of equations:
ix1 +
2x2 =
9
3x1 + (1 + i)x2 = 5 + 8i
x1 = i, x2 = 5
Find the inverse of the matrix
i 1
2 3i
−3
5
2
5
i
1
5
− 15 i
Exponentials
What to do when i is the power of a function?
Exponentials
What to do when i is the power of a function?
Maclaurin (Taylor) Series:
ex = 1 + x +
x2 x3 x4 x5 x6
+
+
+
+
+ ···
2!
3!
4!
5!
6!
Exponentials
What to do when i is the power of a function?
Maclaurin (Taylor) Series:
ex = 1 + x +
e ix = 1 + ix +
x2 x3 x4 x5 x6
+
+
+
+
+ ···
2!
3!
4!
5!
6!
(ix)2 (ix)3 (ix)4 (ix)5 (ix)6
+
+
+
+
+ ···
2!
3!
4!
5!
6!
Exponentials
What to do when i is the power of a function?
Maclaurin (Taylor) Series:
ex = 1 + x +
x2 x3 x4 x5 x6
+
+
+
+
+ ···
2!
3!
4!
5!
6!
(ix)2 (ix)3 (ix)4 (ix)5 (ix)6
+
+
+
+
+ ···
2!
3!
4!
5!
6!
x2
x3 x4
x5 x6
= 1 + ix −
−i
+
+i
−
···
2!
3!
4!
5!
6!
e ix = 1 + ix +
Exponentials
What to do when i is the power of a function?
Maclaurin (Taylor) Series:
ex = 1 + x +
x2 x3 x4 x5 x6
+
+
+
+
+ ···
2!
3!
4!
5!
6!
(ix)2 (ix)3 (ix)4 (ix)5 (ix)6
+
+
+
+
+ ···
2!
3!
4!
5!
6!
x2
x3 x4
x5 x6
= 1 + ix −
−i
+
+i
−
···
2!
3!
4! 5! 6!
x2 x4 x6
x3 x5
= 1−
+
−
+ ··· + i x −
+
− ···
2!
4!
6!
3!
5!
e ix = 1 + ix +
Exponentials
What to do when i is the power of a function?
Maclaurin (Taylor) Series:
x2 x3 x4 x5 x6
+
+
+
+
+ ···
2!
3!
4!
5!
6!
x5 x7
+
−
+ ···
5!
7!
x4 x6
+
−
+ ···
4!
6!
ex = 1 + x +
x3
3!
x2
cos(x) = 1 −
2!
sin(x) = x −
(ix)2 (ix)3 (ix)4 (ix)5 (ix)6
+
+
+
+
+ ···
2!
3!
4!
5!
6!
x2
x3 x4
x5 x6
= 1 + ix −
−i
+
+i
−
···
2!
3!
4! 5! 6!
x2 x4 x6
x3 x5
= 1−
+
−
+ ··· + i x −
+
− ···
2!
4!
6!
3!
5!
e ix = 1 + ix +
Exponentials
What to do when i is the power of a function?
Maclaurin (Taylor) Series:
x2 x3 x4 x5 x6
+
+
+
+
+ ···
2!
3!
4!
5!
6!
x5 x7
+
−
+ ···
5!
7!
x4 x6
+
−
+ ···
4!
6!
ex = 1 + x +
x3
3!
x2
cos(x) = 1 −
2!
sin(x) = x −
(ix)2 (ix)3 (ix)4 (ix)5 (ix)6
+
+
+
+
+ ···
2!
3!
4!
5!
6!
x2
x3 x4
x5 x6
= 1 + ix −
−i
+
+i
−
···
2!
3!
4! 5! 6!
x2 x4 x6
x3 x5
= 1−
+
−
+ ··· + i x −
+
− ···
2!
4!
6!
3!
5!
= cos x + isin x
e ix = 1 + ix +
Does that even make sense?
e ix = cos x + i sin x
Does that even make sense?
e ix = cos x + i sin x
d
ax
dx [e ]
= ae ax ;
Does that even make sense?
e ix = cos x + i sin x
d
ax
dx [e ]
d
ix
dx [e ]
= ae ax ;
Does that even make sense?
e ix = cos x + i sin x
d
ax
ax
dx [e ] = ae ;
d
d
ix
dx [e ]= dx [cos x
+ i sin x]
Does that even make sense?
e ix = cos x + i sin x
d
ax
ax
dx [e ] = ae ;
d
d
ix
dx [e ]= dx [cos x
+ i sin x]
= − sin x + i cos x = i 2 sin x + i cos x = i(cos x + i sin x) = ie ix
Does that even make sense?
e ix = cos x + i sin x
d
ax
ax
dx [e ] = ae ;
d
d
ix
dx [e ]= dx [cos x
+ i sin x]
= − sin x + i cos x = i 2 sin x + i cos x = i(cos x + i sin x) = ie ix
e x+y = e x e y ;
Does that even make sense?
e ix = cos x + i sin x
d
ax
ax
dx [e ] = ae ;
d
d
ix
dx [e ]= dx [cos x
+ i sin x]
= − sin x + i cos x = i 2 sin x + i cos x = i(cos x + i sin x) = ie ix
e x+y = e x e y ;
e ix+iy =
Does that even make sense?
e ix = cos x + i sin x
d
ax
ax
dx [e ] = ae ;
d
d
ix
dx [e ]= dx [cos x
+ i sin x]
= − sin x + i cos x = i 2 sin x + i cos x = i(cos x + i sin x) = ie ix
e x+y = e x e y ;
e ix+iy =e i (x + y ) = cos(x + y ) + i sin(x + y )
= cos x cos y − sin x sin y + i[sin x cos y + cos x sin y ]
= (cos x + i sin y )(cos y + i sin x) = e ix e iy
Does that even make sense?
e ix = cos x + i sin x
d
ax
ax
dx [e ] = ae ;
d
d
ix
dx [e ]= dx [cos x
+ i sin x]
= − sin x + i cos x = i 2 sin x + i cos x = i(cos x + i sin x) = ie ix
e x+y = e x e y ;
e ix+iy =e i (x + y ) = cos(x + y ) + i sin(x + y )
= cos x cos y − sin x sin y + i[sin x cos y + cos x sin y ]
= (cos x + i sin y )(cos y + i sin x) = e ix e iy
Simplify:
e 2+i
π
e 2+ 4 i
e πi + 1
Complex exponentiation: e ix = cos x + i sin x
True or False?
(1) e i = cos 1 + i sin 1
(2) e x = cos x
(3) e ix = e i(x+2π)
(4) e ix = −e i(x+π)
(5) e ix + e −ix is a real number
Coordinates Revisited
Im
Real
Coordinates Revisited
Im
r
θ
Real
Coordinates Revisited
Im
r sin θ
r
θ
r cos θ
Real
Coordinates Revisited
Im
(r cos θ, r sin θ)
r sin θ
r
θ
r cos θ
Real
Coordinates Revisited
Im
(r cos θ, r sin θ)
r sin θ
r
θ
Real
r cos θ
Complex number : r (cos θ + i sin θ) = re iθ
Coordinates Revisited
Im
se iφ
re iθ
Real
Coordinates Revisited
Im
se iφ
re iθ
Real
re iθ · se iφ = (rs)e i(θ+φ)
Coordinates Revisited
Im
se iφ
re iθ
rse i(θ+φ)
Real
re iθ · se iφ = (rs)e i(θ+φ)
Roots of Unity
Im
1
Real
Roots of Unity
Im
1
re iθ
3
=1
Real
Roots of Unity
Im
ei
2π
3
1
ei
4π
3
re iθ
3
=1
Real
Roots of Unity
Im
1
re iθ
5
=1
Real
Roots of Unity
Im
ei
ei
2π
5
4π
5
1
ei
6π
5
ei
re iθ
5
8π
5
=1
Real
Roots of Unity
Im
1
re iθ
12
=1
Real
Roots of Unity
Im
6π
e
ei
ei
i 8π
12
e i 12
4π
e i 12
10π
12
2π
e i 12
1
12π
12
ei
14π
12
ei
ei
16π
12
e
i 18π
12
re iθ
ei
12
20π
12
=1
22π
12
Real
Roots
Find all complex numbers z such that z 3 = 8.
iπ
Find all complex numbers z such that z 3 = 27e 2 .
Find all complex numbers z such that z 4 = 81e 2i .